Answer: A

Small Hint:

Let the original diameter be $d$, so each small semicircle has diameter $\frac{d}{n}$

Big Hint:

Multiply the arc length of one small semicircle by the number $n$ of parts

Solution:

Each small semicircle has diameter $\frac{d}{n},$ so its arc length is $\frac{\pi d}{2n}.$ The sum of all $n$ arc lengths is $$n\cdot\frac{\pi d}{2n}=\frac{\pi d}{2},$$ exactly the semi-circumference of the original circle. This equality holds for every positive $n,$ not merely in the limit.

Thus, the correct answer is A.